WASHINGTON— President Donald Trump, whose modulated tone in addressing Congress Tuesday drew praise mostly from the GOP, now faces the daunting task of lining up votes for a tax overhaul, a replacement for the Affordable Care Act and other agenda items that have splintered his fellow Republicans.

“We are here to start the process,” Mr. Trump said Wednesday, as he met over lunch to plot legislative strategy with top advisers and congressional Republican leaders. “It begins as of now, and we’ll have tremendous success.”